Absolutely Convergent, Conditionally Convergent, or Divergent?

[belvol16]

Homework Statement

∞
Σ (-1)^n-1 n/n² +4
n=1

Homework Equations

lim |a_n+1/a_n| = L
n→∞
b_n+1≤b_n
lim b_n = 0
n→∞

The Attempt at a Solution

So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo.
I found that the ratio test did not work because the limit equaled 1.
Then I used the comparison test and found that the series was convergent...which I'm really confused about because one can compare n/n² +4 to n/n² which is 1/n and is divergent.
Then I compared values of the function to its absolute function and the values were the same...but I'm not sure what that means...
I'm also really not sure what the difference is between being absolutely convergent vs conditionally convergent.
Any help is greatly appreciated.

 

[Ray Vickson]

Homework Statement

∞
Σ (-1)^n-1 n/n² +4
n=1

Homework Equations

lim |a_n+1/a_n| = L
n→∞
b_n+1≤b_n
lim b_n = 0
n→∞

The Attempt at a Solution

So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo.
I found that the ratio test did not work because the limit equaled 1.
Then I used the comparison test and found that the series was convergent...which I'm really confused about because one can compare n/n² +4 to n/n² which is 1/n and is divergent.
Then I compared values of the function to its absolute function and the values were the same...but I'm not sure what that means...
I'm also really not sure what the difference is between being absolutely convergent vs conditionally convergent.
Any help is greatly appreciated.

If you mean what you wrote, the series is obviously divergent, since the nth term is
$$t_n = (-1)^{n-1}\frac{n}{n^2} + 4$$
(according to you) and $\sum_{n=1}^{\infty} 4$ is divergent.
However, if you meant to write
$$t_n = (-1)^{n-1} \frac{n}{n^2+4},$$
then that is a different matter entirely.

Use parentheses to clarify the meaning: a/b+c means $\frac{a}{b} + c$, while a/(b+c) means $\frac{a}{b+c}$.

 

[belvol16]

I mean t_n=(-1)^n-1 * n/(n² +4)
I am new to the fourm and am still learning how to format things.